In the second counterfactual experiment I introduce a conditional cash transfer (CCT) program, which is similar to the design of PROGRESA to both the baseline risk-sharing model and the autarky model. Actual PROGRESA beneficiaries were selected based on multidimensional household characteristics, such as the main materials of walls and floors of household dwellings and ownership of durable goods and assets. Because my model does not have these variables I cannot apply the same criteria. Instead, I choose the low- income type households in the model as beneficiaries. To make the exercise compatible with the actual policy, I take the actual PROGRESA educational grant schedule and con- vert it into adult-equivalent terms and adjust for the economies of scale as was done for
23Under these assumptions, and with the discount factor of 0.95 and risk aversion coefficient of 0.63, the
lifetime utility of a new household is given by:
(lifetime utility) = 120
∑
t=1 δt−1 C1−η−1 1−η =1−δ 120 1−δ C1−η−1 1−η =19.08 C0.37−1 0.37 ,and the consumption-equivalent term,C, of the given lifetime utility is calculated by the following formula:
C= ( lifetime utility)0.37 19.08 +1 0.137 .
Table 18: Educational grant schedule used in counterfactual experiment, 12≤age≤18 School level Grade Monthly grant in pesos
Primary 3 145.83 4 168.48 5 211.22 6 269.15 Secondary 1 410.69 2 403.86 3 457.14
Table 19: Educational grant schedule used in counterfactual experiment, 6≤age≤11 Age Monthly grant in pesos
6 0 7 0 8 145.83 9 145.81 10 170.14 11 206.50
household income and consumption.24 Because households vary in their size, the ad- justed amount varies across households. I take the median value of the adjusted amount of program transfers in each grade level and introduce the amount to the model. Because I do not track the school grade of children before age 12 in the model, for that age group it is not feasible to implement the subsidy schedule based on their completed school grade. Instead, I construct the subsidy schedule based on their age instead of school grade. The constructed semi-annual educational grant schedule is provided in Tables 18 and 19. Ta- ble 20-Table 23 provide the results of the CCT experiments.
The effect of the CCT on schooling outcomes under risk sharing is shown in Table
24 To do this, I first assign the program transfers as shown in Table 1 to the sample households. The
amount of the program transfers assigned to each household corresponds to the completed school grade of the oldest child of that household. Then I rescale the amount of program transfers for each household by an adult-equivalent household size and adjust for the economies of scale.
20. The fraction of children attending school between ages 12 and 18 among benefi- ciary households increases by 0.24, from 0.51 to 0.75, and their completed grade at age 19 increases by 1.17 years, from 7.11 to 8.28. There is little change for non-beneficiary households. There is no change in the outside option value for the non-beneficiary house- holds and their demand for risk sharing is not affected by the introduction of the CCT. Also, the current model assumes that a village is populated by a continuum of house- holds. Because idiosyncratic risks can be pooled among each type and any difference in permanent or deterministic components in income are not shared, the high-income type households are not much affected by changes among the low-income type households. Note that under this experiment households fully anticipate the presence of subsidy from the beginning of their lifecycle. Although there is an equilibrium effect, the effect turns out to be very small on the high-income type (non-beneficiary) households.25
Table 20: The effect of CCT on schooling outcome under risk sharing
No CCT CCT
All Beneficiary All Beneficiary Fraction attending school 0.55 0.51 0.73 0.75 Completed grade at age 19 7.53 7.11 8.37 8.28
Table 21: The effect of CCT on income, consumption, and welfare outcome under risk sharing
No CCT CCT Household consumption 2,634.46 2,661.46 Beneficiary household consumption 1,667.29 1,839.44 Correlation of income and consumption 0.10 0.12 Welfare in consumption equivalence 1,926.66 1,952.93
CCT expenditure - 28.43
Notes: The unit of the reported numbers is Mexican peso.
25Among high-income type households, the fraction attending school slightly increases from 0.6685 to
Table 22: The effect of CCT on schooling outcome under autarky
No CCT CCT
All Beneficiary All Beneficiary Fraction attending school 0.44 0.39 0.69 0.74 Completed grade at age 19 7.10 6.68 8.22 8.26
Table 21 shows the effect of the CCT on household income, consumption, and wel- fare. The reported outcomes are at the village-aggregate level which includes both bene- ficiary and non-beneficiary households. The household consumption level with the CCT is higher by 27 pesos, and the welfare in consumption equivalence increases by 26.27 pe- sos. The correlation between household income and consumption increases by 20 percent, from 0.10 to 0.12. This is because CCT increases the outside option value of beneficiary households which makes(V P)of these households bind more. The reported CCT expen- diture is total CCT amount claimed by beneficiary households in all villages divided by the total measure of households. The CCT expenditure computed in this way is 28.43 pesos, which means that the CCT budget was large enough to give 28.43 pesos to every household in the village in each period. The return from the CCT is calculated as the change in the outcome of interests divided by the CCT expenditure. The computed return is 0.95 for household consumption and 0.92 for welfare.
The effect of the CCT on schooling outcomes under autarky is shown in Table 22. Because there is no interaction among households, non-beneficiary households are not affected by the introduction of the CCT. The fraction of children attending school among beneficiary households increases by 0.35, from 0.39 to 0.74. Their completed grade at age 19 increases by 1.58 years, from 6.68 to 8.26. Household consumption and welfare in consumption equivalence increases by 37.10 and 34.36 pesos respectively. The CCT expenditure is 28.05 pesos. The return from the CCT is computed as in under risk sharing, and it is 1.32 and 1.22 respectively for household consumption and welfare.
Table 23: The effect of CCT on income, consumption, and welfare outcome under autarky No CCT CCT
Household consumption 2,629.79 2,666.89 Beneficiary household consumption 1,802.63 2,025.85 Welfare in consumption equivalence 1,770.81 1,805.17
CCT expenditure - 28.05
Notes: The unit of the reported numbers is Mexican peso.
In comparison to the outcomes under autarky, the effects of the CCT under risk shar- ing are smaller. Under risk sharing, without the CCT, the fraction of children attending school among beneficiary households is 12 percentage points higher under risk sharing than under autarky; however, with the CCT, the difference is only one percentage point. The returns from the CCT in terms of household consumption and welfare are much larger under autarky. There are two possible explanations. First, the CCT also serves as insurance under autarky. Without the CCT, there are more borrowing-constrained house- holds under autarky than under risk sharing. The CCT not only increases the return from schooling but also relaxes the borrowing constraints of households which are hit by neg- ative parental income shocks. However, under risk sharing, the latter effect is negligible because households already had informal insurance that was close to complete risk shar- ing. Second, this may be due to the design of the program transfer schedule which gives cash transfer only up to the 9th grade. Because there are more children who are closer to or above the 9th grade under risk sharing than under autarky without CCT, there are more households that would respond to CCT under autarky (“ceiling effect”).
Table 24: (a) Fraction of children attending school and (b) mean household consumption under autarky by cross-sectional parental-income states
(a) Fraction attending school Deviation from the mean log income No CCT CCT Diff
- two s.d. .36 .68 +.32 - one s.d. .41 .69 +.28 zero .44 .70 +.26 + one s.d. .47 .68 +.21 + two s.d. .49 .64 +.15 (b) Mean consumption Deviation from the mean log income No CCT CCT Diff
- two s.d. 623.03 787.74 +165.02
- one s.d. 1002.22 1167.12 +165.39
zero 1827.51 1988.95 +162.32
+ one s.d. 3741.84 3893.40 +152.61
+ two s.d. 7699.53 7833.98 +136.44
Notes: The unit of the reported numbers is Mexican peso.
Table 25: Effect of CCT with completed grade at age 12 fixed at 2nd grade, beneficiary households only
Risk sharing Autarky
No CCT CCT Diff No CCT CCT Diff Fraction attending school 0.69 0.94 0.25 0.65 0.93 0.28 Completed grade at age 19 5.90 7.11 1.21 5.75 7.09 1.34
I provide two pieces of evidence which support that the first effect is present. The first piece of evidence is provided in Table 24. The increase in enrollment rate is much larger among the households that were hit by negative income shocks. This suggests that households with greater borrowing constraints were further away from an efficient allocation and responded more strongly to the CCT. The second piece of evidence is obtained by conducting another counterfactual experiment which introduces the same CCT, but under an environment where the cap on the 9th grade does not bind. In this
experiment, I force the completed grade at age 12 to be the 2nd grade for every child. As children continue onward from the 2nd grade, the highest grade level they can complete before turning 19 is 9th grade. The schooling outcome under this experiment is provided in Table 25. Again, the differences between the outcomes with and without the CCT in both the fraction attending school and the completed grade level are smaller under risk sharing than under autarky. The results of the original CCT experiment hold even when there is no ceiling effect. However, the difference in the effect of CCT under risk sharing and autarky is smaller in this experiment than in the original CCT experiment, suggesting that ceiling effect is also present.
Another interesting observation from the two counterfactual experiments is that hav- ing risk sharing is much more effective in improving the welfare of households than having CCT, although the effectiveness of CCT is greater when it comes to improving schooling outcomes alone. The difference in household welfare under risk sharing and autarky is 1,156.85 pesos (1,770.81 vs. 2,926.66 pesos). When CCT is introduced to autarky economy the welfare increases only by 34.36 pesos from 1,770.81 to 1,805.17 pesos.
6
Conclusion
This paper is the first to incorporate school attendance choices into an inter-household risk-sharing model with limited-commitment constraints. This paper develops and struc- turally estimates a dynamic risk-sharing model with limited-commitment constraints and school attendance choices in order to study the effectiveness of informal risk sharing in smoothing consumption and schooling, and to evaluate CCT programs when inter- household transfers are allowed. The model considers a village economy comprised of overlapping generations of heterogeneous households. The model parameters are esti-
mated by the simulated method of moments matching the observed income, consump- tion, and child activity choices predicted by the model to those in data from PROGRESA program in Mexico.
Based on the estimated model, I find that the PROGRESA villages are able to smooth consumption and schooling against idiosyncratic income shocks effectively through inter- household transfers. The amount of consumption and children’s school attendance or labor choices are not significantly affected by idiosyncratic income shocks. In contrast to this, a counterfactual simulation of an economy under autarky shows that without inter- household transfers, children’s school attendance and labor choices as well as the amount of consumption are substantially affected by income shocks. The estimated utility cost of attending school for children whose schooling was interrupted in the past is substantial. Therefore, the effect of transitory negative income shocks on children’s school attendance under autarky accumulates over time, leading to lower schooling outcomes compared to the outcomes under risk sharing. Under autarky, the fraction attending school is 11 percentage points lower than that with risk sharing and the completed school grade by age 19 is higher by 0.47 years.
I also conduct counterfactual policy simulations by introducing a CCT program sim- ilar in design to PROGRESA to the model, both with and without risk sharing. Based on the counterfactual simulations I find that the effect of the CCT program on schooling outcomes and the welfare of households is larger under autarky than under risk sharing. The CCT not only increases net returns from schooling by reducing the opportunity cost of attending school, but also mitigates the effect of negative income shocks on household consumption and school attendance. The former effect of the CCT is common under both risk sharing and autarky. The latter effect of the CCT, however, is negligible under risk sharing because the allocations under risk sharing were close to efficient even without the CCT. Moreover, the simulation results show that, under risk sharing, consumption
volatility increases by 20 percents after the introduction of the CCT because beneficiary households can rely on the CCT benefit if inter-household transfer becomes unavailable, and this weakens their voluntary participation incentive to remain in the risk-sharing ar- rangement. This “crowding-out effect” offsets some of the welfare gains from the CCT under risk sharing. Overall, the simulation outcomes suggest that the benefits of CCT may be inflated if the role of inter-household transfers are not taken into account in an evaluation of CCT programs.
Although consumption risk-sharing models considered in previous studies also cap- ture the crowding-out effect of public transfers, the model developed in this paper is the first to study risk sharing and schooling outcomes jointly under CCT programs which are designed to improve schooling outcomes in particular. This cannot be done without a model that explicitly models school attendance choices. Moreover, introducing school attendance choices to a limited-commitment risk-sharing model adds substantial compu- tational challenges, which I overcome by adopting novel theoretical results and computa- tional algorithms (Clausen and Strub, 2013; Fella, 2013). This is an equilibrium model, and the model is simplified in several dimensions (e.g., with a single-child assumption) to keep computational time manageable. Once computation of the model becomes more tractable, developing a richer model that allows for more realistic evaluation of CCT pro- grams in the presence of inter-household transfers would be possible.
There are a number of avenues for future research. First, this paper studies the effect of an existing program design. Ex-ante evaluation of the alternative CCT program designs in the presence of inter-household transfers will also be of interest to both researchers and policymakers. Second, the model considered here assumes a continuum of households, and thus the analysis was confined to large villages. I plan to conducting similar anal- ysis for small villages with a model that assumes a finite number of households as was considered in Ligon, Thomas, and Worrall (2002), Laczó (2008), and Morten (2013).
A
Autarky Problem
I set up an autarky problem for five distinctive lifecycle stages: before a birth occurs (Stage 1), after birth and before a child is aged 12 (Stage 2), while the child is aged be- tween 12 and 18 (Stage 3), after the child finishes schooling and before a migration occurs (Stage 4), and finally, after a migration occurs (Stage 5). In autarky, households maximize their present discounted value of lifetime utility given their period budget constraint,
ci,t=yhi,t
There is no saving or borrowing. Autarky problem is presented in functional equations, and autarky value functions will be denoted byULAut,S whereL=1, ...,5 is a subscript for a lifecycle stage andS∈ f all,springis a subscript for a semester. Households in Stage 3 make a child activity choice. Households in other stages simply consumeyhi,t. Parental type is invariant throughout the lifecycle of a household and is omitted from the equations.
• In Stage 5 households’ lifetime utility under autarky is given by,
U5Aut,f all(si,t) =u(yhi,t) +δEst+1 h U5Aut,spring(si,t+1) i U5Aut,spring(si,t) =u(yih,t) +δ(1−πd)Est+1 h U5Aut,f all(si,t+1) i • In Stage 4,
U4Aut,f all(si,t,Xic,ctype) =u(yhi,t) +δEst+1
h
U4Aut,spring(si,t+1,Xic,ctype)
U4Aut,spring(si,t,Xic,µic) =u(yhi,t) +δ n (1−πm)Est+1|migrate=0 h U4Aut,f all(si,t+1,Xic,µic) i + πmEst+1|migrate=1 h U5Aut,f all(si,t+1) io
wherectypeis a child type.
• In Stage 3, when the age of a child isa,
U3Aut,a,f all(si,t,dischool,t−1 ,Xic,t) =max di,t
n
u(yhi,t) +δEst+1
h
U3Aut,a,spring(si,t+1,dischool,t ,Xic,t)
io
and in the spring, ifa<18,
U3Aut,a,spring(si,t,dischool,t−1 ,Xic,t) =
max
di,t
n
u(yhi,t) +δEst+1
h
U3Aut,a+1,f all(si,t+1,dischool,t ,Xic,t+1)
io
and ifa=18,
U3Aut,a=18,spring(si,t,dischool,t−1 ,Xic,t) =
max
di,t
n
u(yhi,t) +δEst+1
h
U4Aut,f all(si,t+1,Xic,t+1,ctype)
io
• In Stage 2,
U2Aut,a,f all(si,t) =u(yhi,t) +δEst+1
h
U2Aut,a,spring(si,t+1)i
and in the spring, ifa<11,
U2Aut,a,spring(si,t) =u(yhi,t) +δEst+1
h
U2Aut,a,f all(si,t+1)
and ifa=11,
U2Aut,a,spring(si,t) =u(yhi,t) +δEst+1
h
U3Aut,a+1,f all(si,t+1,dischool,t =1,Xic,t+1)i
• In Stage 1, U1Aut,f all(si,t) =u(yhi,t) +δEst+1 h U1Aut,spring(si,t+1) i U1Aut,spring(si,t) =u(yhi,t) +δ n (1−πb)Est+1|birth=0 h U1Aut,f all(si,t+1) i + πbEst+1|birth=1 h U0Aut,2,f all(si,t+1) io
B
Solution Algorithm for the Agent’s Problem
In this subsection, I describe the numerical solution algorithm for the agent’s problem. For the computation, the supports of εp, εh, εw, and ξ are discretized. The elements
in the discretized supports will be denoted by ¯εp, ¯εh, ¯εw, and ¯ξ in the remainder of this
section. The minimum and the maximum value of the discretized support of each shock variable are given by−3σ and+3σ respectively whereσ is the standard deviation of a
given variable. To obtain the right-hand side of (V P) constraints, the autarky problems are solved first. Given the autarky value functions, the agent’s problem is solved by the following procedure.
1. Discretize the support of promised utilityω for each state,Ω:
(a) The lower bound ofω grid is given by the corresponding autarky value
(b) The upper bound is given by the discounted lifetime utility associated with household consumption equivalent to the largest possible yh in every period and the largest possible ¯ξ andXcvalues
(c) A uniform grid of logω with the given lower and upper bounds is constructed.
2. Choose an initial guess ofR:26
(a) The lower bound ofRis given by(1−πm) 1 2
(b) The upper bound ofRis given byδ(1−1
πd).
3. Given R, solve the agent’s problem for Stage 4 and 5. Iterate value functions until they converge. This is a direct application of Krueger and Perri (2011).
4. GivenV4,f all(·), solve Stage 3 problems by using backward induction starting from the spring of age 18.27
(a) Given each choice ofdt, solve the following problem: